Log base 2

What are Logarithmic Functions?

  • A logarithmic function is a function that is the inverse of an exponential function.

  • The purpose of the logarithm is to tell us about the exponent.

  • Logarithmic functions are used to explore the properties of exponential functions and are used to solve various exponential equations.


Representation of a Logarithm Function


\[lo{g_a}b{\text{ }} = x,{\text{ }}then{\text{ }}{a^x} = b\]


What is Log Base 2 or Binary Logarithm?

  • Log base 2 is also known as binary logarithm.

  • It is denoted as (log2n).

  • Log base 2 or binary logarithm is the logarithm to the base 2.

  • It is the inverse function for the power of two functions.

  • Binary logarithm is the power to which the number 2 must be raised in order to obtain the value of n.

  • Here’s the general form.

\[{\mathbf{x}}{\text{ }} = {\text{ }}{\mathbf{lo}}{{\mathbf{g}}_{\mathbf{2}}}{\mathbf{n}}\;\;\;\boxed{} -  -  -  -  -  -  -  -  - \boxed{}{{\mathbf{2}}^{\mathbf{x}}} = {\text{ }}{\mathbf{n}}\]


Graph for Log Base 2

Properties of Log Base 2

There are a few properties of logarithm functions with base 2. They are listed in the table below.


Since we are discussing log base2, we will consider the base to be 2 here.


Basic Log Rules

Product Rule –

 

If the logarithm is given as a product of two numerals, then we can represent the logarithm as the addition of the logarithm of each of the numerals and vice versa.

\[lo{g_b}\left( {x{\text{ }} \times {\text{ }}y} \right){\text{ }} = {\text{ }}lo{g_b}x{\text{ }} + {\text{ }}lo{g_b}y\]

Quotient Rule –


If the logarithm is given as a ratio of two quantities, then it can be written as the difference of the logarithm of each of the numerals.

\[lo{g_b}\left( {\frac{x}{y}} \right)\; = {\text{ }}lo{g_b}x{\text{ }} - {\text{ }}lo{g_b}y\]

Power Rule -


If the logarithm is given in exponential form, then it can be written as exponent times the logarithm of the base.

\[logb({x^k}) = k{\text{ }}lo{g_b}x\]

Zero Rule –


If b is greater than 0, but not equal to 1. The logarithm of x= 1 can be written as 0.

\[lo{g_b}\left( 1 \right){\text{ }} = {\text{ }}0\]

Identity Rule –


When the value of the base b and the argument of the logarithm (inside the parenthesis) are equal then,

\[lo{g_b}\left( b \right){\text{ }} = {\text{ }}1\]

Log of Exponent Rule –


If the base of the exponent is equal to the base of the log then the logarithm of the exponential number is equal to the exponent.

\[lo{g_b}\left( {{b^k}} \right){\text{ }} = {\text{ }}k\]

Exponent of Log Rule –


Raising the logarithm of a number to its base is equal to the number.

\[{b^{logb\left( k \right)}} = k\]


Here are a Few Examples That Show How the Above Basic Rules Work

Example 1 –

Log 40, which can be further written as, \[Log{\text{ }}\left( {20 \times {\text{ }}2} \right)\] by-product rule 

\[ = {\text{ }}log{\text{ }}20{\text{ }} + {\text{ }}log{\text{ }}2\] which is equal to log 40


Example 2 – 

Find the value of log4(4)?

 logb(b) = 1, by identity rule

Therefore, log4(4) = 1.


The Formula for Change of Base

The logarithm can be in the form of log base e or log base 10 or any other bases. Here’s the general formula for change of base -

\[{\mathbf{Lo}}{{\mathbf{g}}_{\mathbf{b}}}\;{\mathbf{x}}{\text{ }} = {\text{ }}\frac{{{\mathbf{Lo}}{{\mathbf{g}}_{\mathbf{a}}}\;{\mathbf{x}}}}{{{\mathbf{Lo}}{{\mathbf{g}}_{\mathbf{a}}}\;{\mathbf{b}}}}\;\]

To find the value of log base 2, we first need to convert it into log base 10 which is also known as a common logarithm.


\[Log{\text{ }}base{\text{ }}2{\text{ }}of{\text{ }}x = \frac{{ln\left( x \right)}}{{ln\left( 2 \right)}}\]


Now you Might Think what Common Logarithmic Function is?

  • Common Logarithmic Function or Common logarithm is the logarithm with base equal to 10.

  • It is also known as the decimal logarithm because of its base.

  • The common logarithm of x is denoted as log x.

  • Example: log 100 = 2 (Since 102= 100).


How to Calculate Log Base 2?

This is how to find log base 2 -

  • According to the log rule,

Log Rule - 

\[log_{b}(x) = y\]

\[b^{y} = x\]

  •  Suppose we have a question, log216 = x

  • Using the log rule,

  • \[\;{2^x} = {\text{ }}16\]

  • We know that 16 in powers of 2 can be written as \[\left( {2 \times 2 \times 2 \times 2{\text{ }} = 16} \right){\text{ }},{2^x} = {2^4}\]

  • Therefore, x is equal to 4.


Questions to be Solved –

Question 1) Calculate the value of log base 2 of 64.

Solution) Here, 

X= 64

Using the formula, \[Log{\text{ }}base{\text{ }}2{\text{ }}of{\text{ }}X = \frac{{ln\left( {64} \right)}}{{ln\left( 2 \right)}} = 6\].

Log base 2 of 64 =\[\frac{{ln\left( {64} \right)}}{{ln\left( 2 \right)}} = 6\].

Therefore, Log base 2 of 64 = 6


Question 2) Find the value of log2(2).

Solution) To find the value of log2(2) we will use the basic identity rule,

\[lo{g_b}\left( b \right){\text{ }} = {\text{ }}1,\].

Therefore, log2(2) = 1.


Question 3) What is the value of log 2 base 10?

Solution) The value of log 2 base 10 can be calculated by the rule,

\[Lo{g_a}\left( b \right){\text{ }} = \frac{{\log b}}{{\log a}}\].

\[Lo{g_{10}}\left( 2 \right){\text{ }} = \frac{{\log 2}}{{\log 10}}\; = {\text{ }}0.3010\].

Therefore, the value of log 2 base 10 = 0.3010.


Question 4) What is the value of log 10 base 2?

Solution) The value of log 10 base 2 can be calculated by the rule,

\[Lo{g_b}\left( a \right){\text{ }} = \frac{{\log b}}{{\log a}}\].

\[Lo{g_2}\left( {10} \right){\text{ }} = \frac{{\log 10}}{{\log 2}}\; = {\text{ 3}}{\text{.3 = 2}}\].

Therefore, the value of log 10 base 2 = 3.32.

FAQ (Frequently Asked Questions)

1. What do you mean by log 2?

In mathematics log 2 is equal to log (2, x). It represents the logarithmic value of x with base equal to 2.


2. How to find log base 2 on a calculator?

 Calculating logarithm with base 2 is an easy task on a calculator. For example, you want to calculate the logarithm base 2 of 8, then these steps show how to find log base 2–

3. What is the value of log 2 Base 2?

The value of log 2 base 2 is equal to 1.

4. How to calculate log base 2 without a calculator?

According to the log rule, this is how to calculate log base 2.

According to the log rule, this is how to calculate log base 2.

 Suppose we have a question, log216 = x

Using the log rule,

2x = 16

We know that 16 in powers of 2 can be written as (2×2×2×2 =16)

2x =24

Therefore, x is equal to 4.